Derive an estimate for the radius of a star

[leonne]

Homework Statement

Consider a star made up of ionized hydrogen.
First, take the electrons to be non-relativistic. If the mass of the star is M, derive
an estimate for the radius R. As the mass increases, how does the radius change

Homework Equations

p~(h² /Me) n^5/3

The Attempt at a Solution

there is a hint saying use "results for the electron gas equations" which is the p~
they got it by P~hⁱme^jn^k than found wht the i j k was in order to get final units in ML^-1 T^-2 I tried doing the same find finding for R so units would be in L but none of the combination worked.

Any hints on this? thanks

 

[mark726]

As the mass of the star increases, the radius will also increase due to the increased gravitational pull. This can be seen from the equation for the radius of a non-relativistic star made up of ionized hydrogen:

R ~ (h^2 / Me)^2/3 * M^(1/3)

As M increases, R will also increase. This can also be seen from the equation for the pressure of an electron gas, which is related to the radius:

P ~ (h^2 / Me) * n^(5/3)

As M increases, the density of the star will also increase (since the mass is increasing but the volume is remaining constant). This means that n will also increase, resulting in a larger pressure and therefore a larger radius.

One way to think about this is that as the mass of the star increases, the gravitational pull also increases, causing the star to compress. This compression leads to a higher density and therefore a larger radius.